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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
P(x) y^{\prime \prime}+Q(x) y^{\prime}+R(x) y=0,\quad x&gt;0.\tag{5.5.1}
\end{equation}
</div>
<p class="continuation">Suppose that <span class="process-math">\(x=0\)</span> is a regular singular point. (Note, if the point is <span class="process-math">\(x=x_0\text{,}\)</span> let <span class="process-math">\(t=x-x_0\text{,}\)</span> then the corresponding point is changed to <span class="process-math">\(t=0\text{.}\)</span> ) Then <span class="process-math">\(x Q(x)/P(x)\)</span> and <span class="process-math">\(x^2 R(x)/P(x)\)</span> are analytic at <span class="process-math">\(x=0\text{.}\)</span> Suppose that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
p(x)=x \frac{Q(x)}{P(x)}=\sum_{n=0}^{\infty} p_n x^n,\quad q(x)=x^2 \frac{R(x)}{P(x)}=\sum_{n=0}^{\infty} q_n x^n,\quad 0&lt;x&lt;\rho,\tag{5.5.2}
\end{equation}
</div>
<p class="continuation">where</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
p_0=\lim \limits_{x \to 0} x \frac{Q(x)}{P(x)},\quad q_0=\lim \limits_{x \to 0} x^2 \frac{R(x)}{P(x)}.
\end{equation*}
</div>
<p class="continuation">Seek a solution in the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
y=\sum_{n=0}^{\infty} a_n x^{n+r}=a_0 x^r+a_1 x^{1+r}+a_2 x^{2+r}+\cdots,\tag{5.5.3}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\(a_0 \neq 0\text{.}\)</span></p>
<span class="incontext"><a href="sec5_5.html#p-230" class="internal">in-context</a></span>
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